Infinite divisibility of the Whittaker distribution

Abstract

In this paper, by using an integral representation of Ismail and Kelker for the quotient of Tricomi hypergeometric functions, we investigate the infinite divisibility and self-decomposability of the recently defined four-parameter lifetime Whittaker distribution, which is a natural extension of the classical gamma, exponential, chi-square, generalized Lindley, Lindley, beta prime, and Lomax distributions. We also show that the Whittaker distribution belongs to the class of hyperbolically completely monotone distributions and generalized gamma convolutions, and it is a super-Gaussian distribution. By using some results for the moments of the Whittaker distribution, we also deduce some Turán type inequalities for the Whittaker functions of the second kind and as an application we show that the effective variance of the Whittaker distribution is bounded from below.